A circle is divided into $ n $ equal parts. Marceline sets out to assign whole numbers from $ 1 $ to $ n $ to each of these pieces so that the distance between two consecutive numbers is always the same. The numbers $ 887 $, $ 217 $ and $ 1556 $ occupy consecutive positions. How many parts was the circumference divided into?

Find all functions $ f: N \rightarrow N $ such that: $\bullet$ $ f (m) = 1 \iff m = 1 $; $\bullet$ If $ d = \gcd (m, n) $, then $ f (mn) = \frac {f (m) f (n)} {f (d)} $; and $\bullet$ $ \forall m \in N $, we have $ f ^ {2012} (m) = m $. Clarification: $f^n (a) = f (f^{n-1} (a))$

In the interior of an equilateral triangle $ ABC $ a point $ P $ is chosen such that $ PA ^2 = PB ^2 + PC ^2 $. Find the measure of $ \angle BPC $.

Given a $ \vartriangle ABC $ with $ AB> AC $ and $ \angle BAC = 60^o$. Denote the circumcenter and orthocenter as $ O $ and $ H $ respectively. We also have that $ OH $ intersects $ AB $ in $ P $ and $ AC $ in $ Q $. Prove that $ PO = HQ $.

Let $ n \in N $. Let's define $ S_n = \{1, ..., n \} $. Let $ x_1 <x_2 <\cdots <x_n $ be any real. Determine the largest possible number of pairs $ (i, j) \in S_n \times S_n $ with $ i \not = j $, for which it is true that $ 1 <| x_i-x_j | <2 $ and justify why said value cannot be higher.